Q.24. Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly

Q: What is the correct answer for JEE Main Mathematics Q24?

The correct answer is A. Sol. Let the center of the circle be $(h,k)$. Since it's in the first quadrant, $h>0, k>0$. The circle intersects the coordinate axes at exactly three points. This means it is tangent to one axis and intersects the other at two points. Since it cuts off equal intercepts from the coordinate axes, it must be tangent to both axes. If it were tangent to one and intersected the other at two points, the

Q: Q.24. Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of…

The correct answer is A. Sol. Let the center of the circle be $(h,k)$. Since it's in the first quadrant, $h>0, k>0$. The circle intersects the coordinate axes at exactly three points. This means it is tangent to one axis and intersects the other at two points. Since it cuts off equal intercepts from the coordinate axes, it must be tangent to both axes. If it were tangent to one and intersected the other at two points, the

Question

Q.24. Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line  is , then the square of the radius of C is

FirstInTest · Accepted Answer

Sol. Let the center of the circle be $(h,k)$. Since it's in the first quadrant, $h>0, k>0$.
The circle intersects the coordinate axes at exactly three points. This means it is tangent to one axis and intersects the other at two points. Since it cuts off equal intercepts from the coordinate axes, it must be tangent to both axes. If it were tangent to one and intersected the other at two points, the intercepts would not be equal unless the center was on $y=x$ or $y=-x$, and the radius was such that it was tangent to one axis and passed through the origin, which would mean only two intersection points. For three points, it must be tangent to one axis and intersect the other at two points, with one of the intersection points being the origin. This would mean unequal intercepts.

However, the problem states "cut off equal intercepts from the coordinate axes". This usually implies that the segments cut by the circle on the x-axis and y-axis are of equal length. If the circle is tangent to both axes, its center is $(r,r)$ and its equation is $(x-r)^2 + (y-r)^2 = r^2$. This circle intersects the axes at $(r,0)$ (tangent to x-axis) and $(0,r)$ (tangent to y-axis). This gives only two intersection points. This contradicts "exactly three points".

Let's reconsider "intersect the coordinate axes at exactly three points". This means the circle passes through the origin $(0,0)$ and is tangent to one of the axes. Or it intersects one axis at two points and is tangent to the other at one point (not the origin). If it passes through the origin, and its center is $(h,k)$ in the first quadrant, then $h^2+k^2=r^2$. If it's tangent to the x-axis, $k=r$. Then $h^2+r^2=r^2 \Rightarrow h=0$, which means the center is on the y-axis, not in the first quadrant. So it cannot pass through the origin and be tangent to an axis while its center is in the first quadrant.

Let's assume the standard interpretation of "intersects the coordinate axes at exactly three points" means it is tangent to one axis and intersects the other at two distinct points. For example, tangent to y-axis at $(0,k)$ and intersects x-axis at $(x_1,0)$ and $(x_2,0)$. The center is $(h,k)$. Then $h=r$. The equation is $(x-r)^2 + (y-k)^2 = r^2$. Intercepts on x-axis are $x^2-2rx+r^2+(0-k)^2=r^2 \Rightarrow x^2-2rx+k^2=0$. The length of the intercept is $|x_1-x_2| = \sqrt{(x_1+x_2)^2-4x_1x_2} = \sqrt{(2r)^2-4k^2} = 2\sqrt{r^2-k^2}$. Intercept on y-axis is 0 (tangent).
This contradicts "equal intercepts from the coordinate axes".

Let's consider the solution's diagram. It shows a circle tangent to the x-axis and intersecting the y-axis at two points. This would mean 3 intersection points. The center is $(h,r)$. The equation is $(x-h)^2 + (y-r)^2 = r^2$. Intercept on x-axis is $(h,0)$. Intercepts on y-axis are $y^2-2ry+h^2=0$. Length of y-intercept is $2\sqrt{r^2-h^2}$.

If the intercepts are equal, then $0 = 2\sqrt{r^2-h^2} \Rightarrow r=h$. This means the circle is tangent to both axes. This leads to only two intersection points $(r,0)$ and $(0,r)$.

There is a contradiction in the problem statement: "exactly three points" and "equal intercepts from the coordinate axes".

Let's assume the diagram is the correct interpretation: tangent to x-axis, intersects y-axis at two points. Center $(h,r)$. $h>0, r>0$. This gives 3 intersection points. The x-intercept is a single point $(h,0)$. The y-intercepts are $(0, y_1)$ and $(0, y_2)$. The length of the x-intercept is 0. The length of the y-intercept is $2\sqrt{r^2-h^2}$. For these to be "equal intercepts", it must mean both are 0. This implies $r=h$ and the circle passes through the origin, which means it's tangent to both axes at the origin, which is not possible for a circle with center in the first quadrant.

Let's assume the problem means the distance from the origin to the points of intersection are equal. This is also not standard.

Let's assume the solution's diagram is correct and it implies the circle is tangent to the x-axis and intersects the y-axis at two points. The center is $(h,r)$. The equation is $(x-h)^2 + (y-r)^2 = r^2$. The x-intercept is $(h,0)$. The y-intercepts are $y^2-2ry+h^2=0$. The length of the chord on the y-axis is $2\sqrt{r^2-h^2}$. The problem states "equal intercepts from the coordinate axes". This is the confusing part. If it means the length of the x-intercept (which is 0 for a tangent point) is equal to the length of the y-intercept, then $2\sqrt{r^2-h^2}=0 \Rightarrow r=h$. This means the circle is tangent to both axes. This would give only two intersection points with the axes, $(r,0)$ and $(0,r)$, contradicting "exactly three points".

Given the solution's diagram, it seems to imply that the circle is tangent to the x-axis and intersects the y-axis at two points. The center is $(h,r)$. The condition "equal intercepts from the coordinate axes" is usually interpreted as the lengths of the chords cut on the axes are equal. If it's tangent to the x-axis, the length of the chord on the x-axis is 0. So the length of the chord on the y-axis must also be 0. This means $r=h$. This leads to the contradiction of 3 points.

Let's assume the problem implies that the circle passes through the origin and is tangent to one axis. If it passes through $(0,0)$ and is tangent to the x-axis, its center is $(h,r)$ and $r=k$. So $(x-h)^2+(y-r)^2=r^2$. Since it passes through $(0,0)$, $h^2+r^2=r^2 \Rightarrow h=0$. Center is $(0,r)$, not in the first quadrant. If it passes through $(0,0)$ and is tangent to the y-axis, its center is $(r,k)$ and $r=h$. So $(x-r)^2+(y-k)^2=r^2$. Since it passes through $(0,0)$, $r^2+k^2=r^2 \Rightarrow k=0$. Center is $(r,0)$, not in the first quadrant.

Let's assume the problem means the circle is tangent to both axes, so center is $(r,r)$. This gives 2 intersection points. This is a common problem type, but contradicts "three points".

Let's assume the problem means the circle is tangent to one axis and passes through the origin. This gives 2 points. This is also not 3.

Let's assume the problem means the circle is tangent to one axis and intersects the other at two points, and one of these points is the origin. This would give 3 points. For example, tangent to x-axis, passes through origin. Center $(h,r)$. $r=k$. $(x-h)^2+(y-r)^2=r^2$. Passes through $(0,0) \Rightarrow h^2+r^2=r^2 \Rightarrow h=0$. Center $(0,r)$. Not in first quadrant.

This problem statement is highly ambiguous. Let's try to interpret the solution's diagram. It shows a circle tangent to the x-axis and intersecting the y-axis at two points. The center is $(h,r)$. The x-intercept is $(h,0)$. The y-intercepts are $(0, y_1)$ and $(0, y_2)$. The length of the chord on the y-axis is $2\sqrt{r^2-h^2}$. The problem states "equal intercepts from the coordinate axes". This must mean that the x-intercept is also a chord of length $2\sqrt{r^2-h^2}$. This is only possible if the circle intersects the x-axis at two points, not tangent. This contradicts the diagram.

Let's assume the diagram is misleading and the circle intersects both axes at two points, and the lengths of these chords are equal. Let the center be $(h,k)$. The equation is $(x-h)^2+(y-k)^2=r^2$. The length of the x-intercept is $2\sqrt{r^2-k^2}$. The length of the y-intercept is $2\sqrt{r^2-h^2}$. If these are equal, then $k=h$. So the center is $(h,h)$.
Now, for "exactly three points". If the center is $(h,h)$, and it intersects the axes at two points each, that's 4 points. For 3 points, one of the intersection points must be the origin. If it passes through the origin, then $h^2+h^2=r^2 \Rightarrow 2h^2=r^2$. The circle equation is $(x-h)^2+(y-h)^2=2h^2$. This circle passes through $(0,0)$. It intersects the x-axis at $(0,0)$ and $(2h,0)$. It intersects the y-axis at $(0,0)$ and $(0,2h)$. This gives 3 distinct points: $(0,0), (2h,0), (0,2h)$. This fits "exactly three points".

So, the center is $(h,h)$ and $r^2=2h^2$. The circle equation is $(x-h)^2+(y-h)^2=2h^2$. The center is in the first quadrant, so $h>0$.

Now, the length of the chord of C on the line $x+y=1$ is $\sqrt{14}$.
The line is $x+y-1=0$. The distance from the center $(h,h)$ to this line is $d = \frac{|h+h-1|}{\sqrt{1^2+1^2}} = \frac{|2h-1|}{\sqrt{2}}$.
The length of the chord is $2\sqrt{r^2-d^2}$.
We are given $2\sqrt{r^2-d^2} = \sqrt{14}$.
$4(r^2-d^2) = 14 \Rightarrow 2(r^2-d^2) = 7$.
Substitute $r^2=2h^2$ and $d^2 = \frac{(2h-1)^2}{2}$:
$2(2h^2 - \frac{(2h-1)^2}{2}) = 7$
$4h^2 - (2h-1)^2 = 7$
$4h^2 - (4h^2-4h+1) = 7$
$4h-1 = 7 \Rightarrow 4h = 8 \Rightarrow h=2$.

We need the square of the radius, $r^2$.
$r^2 = 2h^2 = 2(2^2) = 2(4) = 8$.

This interpretation resolves all contradictions and matches the answer. The diagram in the solution is misleading for the "equal intercepts" part, but the final result is consistent with this interpretation.

Answer

8

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2

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4

Answer

1

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