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Chapter 15: Problem 1401

The vertex of the parabola \((x-b)^{2}=4 b(y-b)\) is ........ (a) (b,0) (b) \((0, b)\) (c) \((0,0)\) (d) \((\mathrm{b}, \mathrm{b})\)

Short Answer

Expert verified
The vertex of the given parabola \((x-b)^2 = 4b(y-b)\) is (b, b), which corresponds to option (d).

Step by step solution

01

Identify the vertex form of the given parabolic equation

We have the given equation of the parabola as \((x-b)^2 = 4b(y-b)\). Comparing it with the standard vertex form of a parabola, \((x-h)^2 = 4a(y-k)\), we can match the corresponding values.
02

Compare the given equation to the vertex form equation

We have: \((x-b)^2 = 4b(y-b)\) \((x-h)^2 = 4a(y-k)\) Comparing the two equations, we get: h = b k = b a = b Thus, the vertex of the parabola is at the point (h, k) which is (b, b).
03

Determine the correct answer

According to our finding, the vertex of the given parabola is (b, b). Comparing this with the given options, we see that the vertex of the parabola is the point (b, b) which corresponds to option (d). Therefore, the correct answer is (d) \((b, b)\).

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Most popular questions from this chapter

The equation of the common tangent to the curves \(\mathrm{y}^{2}=8 \mathrm{x}\) and \(\mathrm{xy}=-1\) is (a) \(9 x-3 y+2=0\) (b) \(2 \mathrm{x}-\mathrm{y}+1=0\) (c) \(x-2 y+8=0\) (d) \(x-y+2=0\)

The equation \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}-8 \mathrm{x}-18 \mathrm{y}+35=\mathrm{k}\) represents (a) parabola if \(\mathrm{k}>0\) (b) circle if \(\mathrm{k}>0\) (c) a point if \(\mathrm{k}=0\) (d) a hyperbola if \(\mathrm{k}>0\)

The radius of the circle passing through the points \((5,2)\), \((5,-2)\) and \((1,2)\) is (a) \(2 \sqrt{5}\) (b) \(3 \sqrt{2}\) (c) \(5 \sqrt{2}\) (d) \(2 \sqrt{2}\)

If one of the diameters of the circle \(x^{2}+y^{2}-2 x-6 y+6=0\) is a chord to the circle with centre \((2,1)\), then the radius of the circle is \(\ldots\) (a) 3 (b) \(\sqrt{3}\) (c) 2 (d) \(\sqrt{2}\)

The equations of the common tangents to the parabola \(y=x^{2}\) and \(y=-(x-2)^{2}\) is (a) \(\mathrm{y}=4(\mathrm{x}-1)\) (b) \(\mathrm{y}=2\) (c) \(\mathrm{y}=-4(\mathrm{x}-1)\) (d) \(\mathrm{y}=-30 \mathrm{x}-50\)
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