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Chapter 15: Problem 1384

The lines \(2 \mathrm{x}-3 \mathrm{y}-5=0\) and \(3 \mathrm{x}-4 \mathrm{y}-7=0\) are diameters of a circle of area 154 square units then the equation of the circle is (a) \(x^{2}+y^{2}+2 x-2 y-62=0\) (b) \(x^{2}+y^{2}+2 x-2 y-47=0\) (c) \(x^{2}+y^{2}-2 x+2 y-47=0\) (d) \(x^{2}+y^{2}-2 x+2 y-62=0\)

Short Answer

Expert verified
The equation of the circle is \((x^2 + y^2) + 2x - 2y - 62 = 0\).

Step by step solution

01

Find the radius of the circle

Since the area of the circle is 154 square units, we can determine the radius using the formula: Area = πr² => r² = Area/π = 154/π
02

Find the intersection point of the given lines

To find the center of the circle, we need to solve the system of linear equations formed by the given lines: 1. \(2x - 3y - 5 = 0\) 2. \(3x - 4y - 7 = 0\) Solve these two linear equations to find the values of x and y.
03

Write the equation of the circle

Using the general form of the equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), insert the values obtained for (h, k), and r² from step 1 and step 2: \((x - h)^2 + (y - k)^2 = r^2\) Now, expand this equation and compare it to the given options to find the equation of the circle.

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Most popular questions from this chapter

Let \(E\) be the ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 4\right)=1\) and \(C\) be the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}=9\). Let \(\mathrm{P}\) and \(\mathrm{Q}\) be the point \((1,2)\) and \((2,1)\) respe. Then (a) P lies inside \(\mathrm{C}\) but outside \(\mathrm{E}\) (b) P lies inside both \(\mathrm{C}\) and \(\mathrm{E}\) (c) \(Q\) lies outside both \(\mathrm{C}\) and \(\mathrm{E}\) (d) Q lies inside \(\mathrm{C}\) but outside \(\mathrm{E}\)

The equation \(\mid \sqrt{\left[x^{2}+(y+1)^{2}\right]-\sqrt\left[x^{2}+(y-1)^{2}\right] \mid=k \text { will }}\) represent a hyperbola for (a) \(k \in(0,3)\) (b) \(\mathrm{k} \in(2,3)\) (c) \(k \in(-3,0)\) (d) \(\mathrm{k} \in(0,2)\)

The line \((\mathrm{x}+\mathrm{g}) \cos \theta+(\mathrm{y}+\mathrm{f}) \sin \theta=\mathrm{k}\) touches the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{~g} \mathrm{x}+2 \mathrm{fy}+\mathrm{c}=0\) only its (a) \(g^{2}+f^{2}=c+k^{2}\) (b) \(g^{2}+f^{2}=c^{2}+k^{2}\) (c) \(g^{2}+f^{2}=c-k^{2}\) (d) \(g^{2}+f^{2}=c^{2}-k^{2}\)

The distance from the foci of \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) on the ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 25\right)=1\) is \(\ldots \ldots \ldots\) (a) \(4-(5 / 4) \mathrm{y}_{1}\) (b) \(5-(4 / 5) \mathrm{y}_{1}\) (c) \(5-(4 / 5) \mathrm{x}_{1}\) (d) \(4-(4 / 5) \mathrm{y}_{1}\)

The equation of the chord of parabola \(\mathrm{y}^{2}=8 \mathrm{x}\). Which is bisected at the point \((2,-3)\) is (a) \(3 x+4 y-1=0\) (b) \(4 x+3 y+1=0\) (c) \(3 \mathrm{x}-4 \mathrm{y}+1=0\) (d) \(4 x-3 y-1=0\)
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